I'm wondering if anybody could suggest some techniques that might be brought to bear on the following problem: Suppose a finite sequence [tex]M_1,M_2,\dots,M_k[/tex] of [tex]4\times 4[/tex] orthogonal reflection matrices is given. I'm interested in determining conditions on these matrices that will guarantee that the product of any ordered sub-collection having an even and nonzero number of elements does not have eigenvalue 1. This is really the same as asking whether the result will be a simple or "double" rotation. If it's a simple rotation, the product will have an axis-plane that is fixed point-by-point and so will have eigenvalue 1. If it's a "double" rotation (two independent rotations in orthogonal 2-planes), then it will not have eigenvalue 1. Or better yet: does anybody have any ideas for generating a list of matrices satisfying the specified property (besides just randomly generating matrices and testing all generated collections)?